Gaussian fluctuations of the moderately interacting Shigesada-Kawasaki-Teramoto particle system around the mollified mean-field limit
Abstract
We study, at the logarithmic mollification scale η−2(d+1+α)≤δlog N of Chen, Daus, Holzinger and J{ü}ngel, the fluctuation field VN=√N(μN−uη) of the moderately interacting multi-species particle system whose mean-field limit is the Shigesada- Kawasaki-Teramoto (SKT) cross-diffusion system. The field is centred at the solution uη of the mollified equation, which depends on N through η. On a scale of weighted Bessel-potential spaces of negative order we prove that the laws of (VN,MN) are tight, where MN is the martingale part. Every limit point of MN is a continuous Gaussian martingale with an explicit deterministic bracket. Every limit point of VN is a weak solution, and the mild solution, of a linear Ornstein-Uhlenbeck SPDE whose drift is the linearisation of the SKT operator at the limit u. The initial field and the martingale are independent, and weak solutions are unique given both, so the whole sequence converges in law, jointly with the martingale part, and the limit is unique in law among solutions of the martingale problem. The noise is d Mi=div(√(2uiΘi)dWi), with Θi=σi+∑jf(aijuj) and W a cylindrical Wiener process on L2(ℝd;ℝnd). The hypotheses are those of Chen, Daus, Holzinger and J{ü}ngel, including the smallness and the nonnegativity on ℝ of the nonlinearity f. We add radial kernels Bij, the independence of the initial data and the Brownian motions across particles and species, a smaller δ, a regularity budget on u0, a strengthened smallness condition on f that makes the backward system diagonally dominant in weighted spaces, and an integer Sobolev index s.
Related
- Gaussian fluctuations for the stochastic wave equation with drift
- Gaussian zero-count fluctuations in annuli
- Gaussian fluctuations of differential observables of stationary lattice fields
- Gaussian Fluctuations of the Gradient of the Fréchet Function and Nonstandard Asymptotics of Sample Fréchet Means
- Mollified disc multipliers
BibTeX
@misc{huber2026gaussian,
title = {{Gaussian fluctuations of the moderately interacting Shigesada-Kawasaki-Teramoto particle system around the mollified mean-field limit}},
author = {Florian Huber},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08601},
eprint = {2611.08601},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08601},
}
RIS
TY - UNPB
TI - Gaussian fluctuations of the moderately interacting Shigesada-Kawasaki-Teramoto particle system around the mollified mean-field limit
AU - Florian Huber
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08601
UR - https://arxiv.org/abs/2611.08601
AB - We study, at the logarithmic mollification scale η^−2(d+1+α)≤δlog N of Chen, Daus, Holzinger and J{ü}ngel, the fluctuation field V^N=√N(μ^N−u_η) of the moderately interacting multi-species particle system whose mean-field limit is the Shigesada- Kawasaki-Teramoto (SKT) cross-diffusion system. The field is centred at the solution u_η of the mollified equation, which depends on N through η. On a scale of weighted Bessel-potential spaces of negative order we prove that the laws of (V^N,M^N) are tight, where M^N is the martingale part. Every limit point of M^N is a continuous Gaussian martingale with an explicit deterministic bracket. Every limit point of V^N is a weak solution, and the mild solution, of a linear Ornstein-Uhlenbeck SPDE whose drift is the linearisation of the SKT operator at the limit u. The initial field and the martingale are independent, and weak solutions are unique given both, so the whole sequence converges in law, jointly with the martingale part, and the limit is unique in law among solutions of the martingale problem. The noise is d M_i=div(√(2u_iΘ_i)dW_i), with Θ_i=σ_i+∑_jf(a_iju_j) and W a cylindrical Wiener process on L^2(ℝ^d;ℝ^nd). The hypotheses are those of Chen, Daus, Holzinger and J{ü}ngel, including the smallness and the nonnegativity on ℝ of the nonlinearity f. We add radial kernels B_ij, the independence of the initial data and the Brownian motions across particles and species, a smaller δ, a regularity budget on u_0, a strengthened smallness condition on f that makes the backward system diagonally dominant in weighted spaces, and an integer Sobolev index s.
ER -